MATHEMATICAL PROGRAMMING PROBLEMS 12 hrs

LINEAR PROGRAMMING EXAMPLES AND SOLUTIONS

This file contains four Linear Programming problems examples. Each problem is first

expressed in mathematical notation; next the models are expressed in GAMS notation;

and finally the model is solved and the results are found in the a GAMS LIST file. We will

go over the problems in class and interpret the results. You will then write and execute

GAMS programs to solve the problems assigned to you.

GAMS program is written in TEXT format. You can either write your model in GAMS

notation in the GAMS Editor window or use WORD or NOTEPAD to type the GAMS

commands for the problem you are working on and save the file as a TEXT file and then

paste it into GAMS Editor. Save the files under names which will help you remember as to

what the file contains. For example PROB1 if the file pertains to Problem 1.

When writing your GAMS code, never use TABS and always leave the first position of

each line blank (it is reserved for the special characters). For example, a “*” character in

the first position will allow you to write notes or comments that the program will ignore.

When you are ready to solve the problem by executing the GAMS program, click on the

“RUN” icon. If you made no errors, GAMS will solve the problem and it will save the

results in a list file with the same name but with an extension “LST.” For example

PROB1.LST. If you made errors, GAMS will indicate it on the monitor. Errors are found

by looking at lines the start with **** in the LST file. There will be a $ sign right under

the place where your error has occurred followed by a number. At the end of the LST file,

the error(s) number will be listed followed by an explanation. Once you know your

error(s), go back into your GAMS command file and make corrections. Save the file and

run the program again.

You can download a User’s Guide and a free student copy of GAMS Window version

from the web site http://palette.ecn.purdue.edu/~rardin/gams/notes.html and download

gidev091.exe.

EXAMPLE 1:

A clothes manufacturer makes products Q1: Pants and Q2: Jackets. The net profit from

Pants p1=$5 and from jackets p2=$10. The firm is facing the following time constraints on

the production processes:

Cutting 0.5Q1 + 2.5Q2 = 100 hours

Sawing 1.0Q1 + 0.5Q2 = 70 hours

Packaging 1.0Q1 + 1.0Q2 = 80 hours

Find the profit maximizing levels of Q1 and Q2.

EXAMPLE 1: Model in Mathematical Notation

2 Maximize ( = ' pi*Qi i =1

2 Subject to: ' aij*Qi # bj i= 1,2 and j = 1,2,3 i =1

Where: pi is the net profit and Qi is the output of good i, respectively aij is amount of resource j used per unit of output of good i bj is the total amount of resource j available.

EXAMPLE 1: Model in GAMS Notation

$TITLE DR. KONYAR'S LINEAR PROGRAMMING EXAMPLE 1 $OFFSYMLIST OFFSYMXREF OPTION NLP = MINOS5

SETS I GOODS /JACK, PANT / J PRODUCTION PROCESSES /CUT, SAW, PACK/;

*Data PARAMETERS NETPROF(I) NET PROFIT PER UNIT / JACK 5 PANT 10 /

PRDLEVEL(J) PRODUCTION CONSTRAINT LEVELS / CUT 100, SAW 70, PACK 80 /;

TABLE USEPARAM(J,I) PRODUCTION PROCESS USE PARAMETERS

JACK PANT (Note: When creating data in a TABLE format, at least CUT .5 2.5 one part of the number must be vertically aligned SAW 1 .5 with the column label under which it belongs) PACK 1 1 ;

VARIABLES Q(I) OUTPUT LEVEL PROF TOTAL PROFIT;

POSITIVE VARIABLE Q; (Note: This is akin to the non-negativity constraint)

EQUATIONS OBJFUNC OBJECTIVE FUNCTION CONST(J) CONSTRAINS;

OBJFUNC .. PROF =E= SUM(I, NETPROF(I)*Q(I)); CONST(J) .. SUM(I, Q(I) * USEPARAM(J,I)) =L= PRDLEVEL(J);

MODEL PROFIT /OBJFUNC,CONST/ SOLVE PROFIT USING LP MAXIMIZING PROF;

EXAMPLE 1: GAMS LIST FILE

GAMS 2.25.087 386/486 DOS 02/25/98 14:41:35 PAGE 1

DR. KONYAR'S LINEAR PROGRAMMING EXAMPLE

COMPILATION TIME = 0.050 SECONDS VERID MW2-25-087 Equation Listing SOLVE PROFIT USING LP FROM LINE 53

---- OBJFUNC =E= OBJECTIVE FUNCTION

OBJFUNC.. - 5*Q(JACK) - 10*Q(PANT) + PROF =E= 0 ; (LHS = 0)

---- CONST =L= CONSTRAINS

CONST(CUT).. 0.5*Q(JACK) + 2.5*Q(PANT) =L= 100 ; (LHS = 0)

CONST(SAW).. Q(JACK) + 0.5*Q(PANT) =L= 70 ; (LHS = 0)

CONST(PACK).. Q(JACK) + Q(PANT) =L= 80 ; (LHS = 0)

MODEL STATISTICS

BLOCKS OF EQUATIONS 2 SINGLE EQUATIONS 4 BLOCKS OF VARIABLES 2 SINGLE VARIABLES 3 NON ZERO ELEMENTS 9

GENERATION TIME = 0.050 SECONDS

EXECUTION TIME = 0.170 SECONDS VERID MW2-25-087 Solution Report SOLVE PROFIT USING LP FROM LINE 53

S O L V E S U M M A R Y

MODEL PROFIT OBJECTIVE PROF TYPE LP DIRECTION MAXIMIZE SOLVER MINOS5 FROM LINE 53

**** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 1 OPTIMAL **** OBJECTIVE VALUE 550.0000

RESOURCE USAGE, LIMIT 0.270 5400.000 ITERATION COUNT, LIMIT 2 10000 M I N O S 5.3 (Nov 1990) Ver: 225-386-02 = = = = =

EXIT -- OPTIMAL SOLUTION FOUND

LOWER LEVEL UPPER MARGINAL

---- EQU OBJFUNC . . . 1.000 OBJFUNC OBJECTIVE FUNCTION

---- EQU CONST CONSTRAINS

LOWER LEVEL UPPER MARGINAL

CUT -INF 100.000 100.000 2.500 SAW -INF 65.000 70.000 . PACK -INF 80.000 80.000 3.750

Note: Marginal value is same as the opportunity cost of a scarce, i.e., binding resource. It is also know as dual value, shadow value, imputed value, or accounting value. It measures the amount by which the objective function value would change if the resource constraint was increased by one unit. For example, if the cutting constraint was to be increased by 1 hour then the profits will increase by $2.50. Which means the cutting capacity has a scarcity value of $2.50.

---- VAR Q OUTPUT LEVEL

LOWER LEVEL UPPER MARGINAL

JACK . 50.000 +INF . PANT . 30.000 +INF .

Solution Report SOLVE PROFIT USING LP FROM LINE 53

LOWER LEVEL UPPER MARGINAL

---- VAR PROF -INF 550.000 +INF .

PROF TOTAL PROFIT

**** REPORT SUMMARY : 0 NONOPT 0 INFEASIBLE 0 UNBOUNDED

EXECUTION TIME = 0.160 SECONDS VERID MW2-25-087

USER: U.S. Department of Agriculture G950224:1314CR-MW2 Washington, DC

**** FILE SUMMARY

INPUT C:\COURSE\E480\EXP1.GMS OUTPUT C:\COURSE\E480\EXP1.LST

EXAMPLE 2: Model in Mathematical Notation

2 Minimize C = ' ci*Qi i =1

2 Subject to: ' aij*Qi $ bj i= 1,2 and j = 1,2,3 i =1

Where: ci is the per unit cost of and Qi is the amount of feed i, respectively aij is amount of nutrient j provided by a unit of feed i bj is the minimum amount of nutrient j needed per cattle

EXAMPLE 2: Model in GAMS Notation

$TITLE PETERSEN & LEWIS CHAPTER 8, Page 282

* Constrained Cost Minimization: A farmer is trying to minimize feed cost of feeding a * cow, subject to minimum nutritional requirements of the cow.

$OFFSYMLIST OFFSYMXREF OPTION NLP = MINOS5

SETS I FEEDS / A, B / J NUTRIENTS / PROTEIN, CALCIUM, CARBO /

*Data PARAMETERS COST(I) FEED PRICES / A 100 B 200 /

MINREQ(J) MINIMUM FEED REQUIREMENTS / PROTEIN 40 CALCIUM 60 CARBO 60 /;

TABLE NUTRI(J,I) UNITS OF NUTRIENTS PER TON A B PROTEIN 1 1 CALCIUM 3 1 CARBO 1 6 ;

VARIABLES EAT(I) FEED FEEDING LEVELS TOTCOST TOTAL COST;

POSITIVE VARIABLE EAT;

EQUATIONS OBJFUNC OBJECTIVE FUNCTION CONST(J) CONSTRAINS;

OBJFUNC .. TOTCOST =E= SUM(I, COST(I)*EAT(I)); CONST(J) .. SUM(I, EAT(I)*NUTRI(J,I)) =G= MINREQ(J);

MODEL MINCOST /OBJFUNC,CONST/ SOLVE MINCOST USING LP MINIMIZING TOTCOST;

EXAMPLE 2: GAMS LIST FILE

GAMS 2.25.087 386/486 DOS 02/25/98 14:41:42 PAGE 1 PETERSEN & LEWIS CHAPTER 8, Page 282

COMPILATION TIME = 0.050

Equation Listing SOLVE MINCOST USING LP FROM LINE 57

---- OBJFUNC =E= OBJECTIVE FUNCTION

OBJFUNC.. - 100*EAT(A) - 200*EAT(B) + TOTCOST =E= 0 ; (LHS = 0)

---- CONST =G= CONSTRAINS

CONST(PROTEIN).. EAT(A) + EAT(B) =G= 40 ; (LHS = 0 ***)

CONST(CALCIUM).. 3*EAT(A) + EAT(B) =G= 60 ; (LHS = 0 ***)

CONST(CARBO).. EAT(A) + 6*EAT(B) =G= 60 ; (LHS = 0 ***)

Column Listing SOLVE MINCOST USING LP FROM LINE 57

Model Statistics SOLVE MINCOST USING LP FROM LINE 57

MODEL STATISTICS

BLOCKS OF EQUATIONS 2 SINGLE EQUATIONS 4 BLOCKS OF VARIABLES 2 SINGLE VARIABLES 3 NON ZERO ELEMENTS 9

GENERATION TIME = 0.110 SECONDS

EXECUTION TIME = 0.170

Solution Report SOLVE MINCOST USING LP FROM LINE 57

S O L V E S U M M A R Y

MODEL MINCOST OBJECTIVE TOTCOST TYPE LP DIRECTION MINIMIZE SOLVER MINOS5 FROM LINE 57

**** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 1 OPTIMAL **** OBJECTIVE VALUE 4400.0000

RESOURCE USAGE, LIMIT 0.330 5400.000 ITERATION COUNT, LIMIT 2 10000 EXIT -- OPTIMAL SOLUTION FOUND

LOWER LEVEL UPPER MARGINAL ---- EQU OBJFUNC . . . 1.000

OBJFUNC OBJECTIVE FUNCTION

---- EQU CONST CONSTRAINS

LOWER LEVEL UPPER MARGINAL

PROTEIN 40.000 40.000 +INF 80.000 CALCIUM 60.000 112.000 +INF . CARBO 60.000 60.000 +INF 20.000

---- VAR EAT FEED FEEDING LEVELS

LOWER LEVEL UPPER MARGINAL

A . 36.000 +INF . B . 4.000 +INF

Solution Report SOLVE MINCOST USING LP FROM LINE 57

LOWER LEVEL UPPER MARGINAL

---- VAR TOTCOST -INF 4400.000 +INF .

TOTCOST TOTAL COST

**** REPORT SUMMARY : 0 NONOPT 0 INFEASIBLE 0 UNBOUNDED

EXECUTION TIME = 0.110 SECONDS VERID MW2-25-087

USER: U.S. Department of Agriculture G950224:1314CR-MW2 Washington, DC

**** FILE SUMMARY

INPUT C:\COURSE\E480\EXP2.GMS OUTPUT C:\COURSE\E480\EXP2.LST

EXAMPLE 3: Model in Mathematical Notation

2 3 Minimize C = ' ' cij*Qij i=1 j=1

3 Subject to: ' Qij # bi j=1

2 and ' Qij $ dj i= 1,2 and j = 1,2,3 i=1

Where: cij is the per unit cost of shipping a car and Qij is the amount of cars shipped from plant i to dealer j, respectively bj is the maximum amount of cars produced at plant i dj is the minimum amount of cars needed at dealer j.

EXAMPLE 3: Model in GAMS Notation

$TITLE PETERSEN & LEWIS CHAPTER 8, Page 287

* Constrained Cost Minimization: An auto manufacturer with plants in Detroit * and Los Angeles, wants to minimize its shipping costs of sending cars to dealers in * Atlanta, Chicago, and Denver, while making sure not to exceed each plant’s capacity * and satisfy the minimum demand for cars from its dealerships.

$OFFSYMLIST OFFSYMXREF OPTION NLP = MINOS5

SETS I PLANTS / DETROIT, LOSANGEL/ J DEALERS / ATLANTA, CHICAGO, DENVER /

*Data PARAMETER SUPPLY(I) NUMBER OF CARS PRODUCED / DETROIT 3000 LOSANGEL 5000 /

DEMAND(J) NUMBER OF CARS DEMANDED

/ ATLANTA 3000 CHICAGO 4000 DENVER 1000 /;

TABLE TRANCOST(I,J) TRANSPORTATION COST PER CAR

ATLANTA CHICAGO DENVER DETROIT 200 100 300 LOSANGEL 400 300 200 ;

VARIABLES SHIPPED(I,J) CARS SHIPPED FROM I TO J TOTCOST TOTAL COST OF SHIPPING CARS;

POSITIVE VARIABLE SHIPPED;

EQUATIONS OBJFUNC OBJECTIVE FUNCTION SUPPLCON(I) SUPPLY CONSTRAINT DEMANCON(J) DEMAND CONSTRAINT;

OBJFUNC .. TOTCOST =E= SUM((I,J), TRANCOST(I,J)*SHIPPED(I,J)); SUPPLCON(I) .. SUM(J, SHIPPED(I,J)) =L= SUPPLY(I); DEMANCON(J) .. SUM(I, SHIPPED(I,J)) =G= DEMAND(J);

MODEL MINCOST / OBJFUNC, SUPPLCON, DEMANCON /; SOLVE MINCOST USING LP MINIMIZING TOTCOST;

EXAMPLE 3: GAMS LIST FILE

GAMS 2.25.087 386/486 DOS 02/25/98 15:44:47 PAGE 1 PETERSEN & LEWIS CHAPTER 8, Page 287 COMPILATION TIME = 0.060

Equation Listing SOLVE MINCOST USING LP FROM LINE 59

---- OBJFUNC =E= OBJECTIVE FUNCTION

OBJFUNC.. - 200*SHIPPED(DETROIT,ATLANTA) - 100*SHIPPED(DETROIT,CHICAGO)

- 300*SHIPPED(DETROIT,DENVER) - 400*SHIPPED(LOSANGEL,ATLANTA)

- 300*SHIPPED(LOSANGEL,CHICAGO) - 200*SHIPPED(LOSANGEL,DENVER)+TOTCOST =E= 0; (LHS = 0)

---- SUPPLCON =L= SUPPLY CONSTRAINT

SUPPLCON(DETROIT).. SHIPPED(DETROIT,ATLANTA) + SHIPPED(DETROIT,CHICAGO)

+ SHIPPED(DETROIT,DENVER) =L= 3000 ; (LHS = 0)

SUPPLCON(LOSANGEL).. SHIPPED(LOSANGEL,ATLANTA) + SHIPPED(LOSANGEL,CHICAGO)

+ SHIPPED(LOSANGEL,DENVER) =L= 5000 ; (LHS = 0)

---- DEMANCON =G= DEMAND CONSTRAINT

DEMANCON(ATLANTA).. SHIPPED(DETROIT,ATLANTA) + SHIPPED(LOSANGEL,ATLANTA) =G= 3000 ; (LHS = 0 ***)

DEMANCON(CHICAGO).. SHIPPED(DETROIT,CHICAGO) + SHIPPED(LOSANGEL,CHICAGO) =G= 4000 ; (LHS = 0 ***)

DEMANCON(DENVER).. SHIPPED(DETROIT,DENVER) + SHIPPED(LOSANGEL,DENVER) =G= 1000 ; (LHS = 0 ***)

Model Statistics SOLVE MINCOST USING LP FROM LINE 59

MODEL STATISTICS

BLOCKS OF EQUATIONS 3 SINGLE EQUATIONS 6 BLOCKS OF VARIABLES 2 SINGLE VARIABLES 7 NON ZERO ELEMENTS 19

GENERATION TIME = 0.060 SECONDS

EXECUTION TIME = 0.110

S O L V E S U M M A R Y

MODEL MINCOST OBJECTIVE TOTCOST TYPE LP DIRECTION MINIMIZE SOLVER MINOS5 FROM LINE 59

**** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 1 OPTIMAL **** OBJECTIVE VALUE 2000000.0000

RESOURCE USAGE, LIMIT 0.220 5400.000 ITERATION COUNT, LIMIT 4 10000 EXIT -- OPTIMAL SOLUTION FOUND

LOWER LEVEL UPPER MARGINAL ---- EQU OBJFUNC . . . 1.000

OBJFUNC OBJECTIVE FUNCTION

---- EQU SUPPLCON SUPPLY CONSTRAINT

LOWER LEVEL UPPER MARGINAL DETROIT -INF 3000.000 3000.000 -200.000 LOSANGEL -INF 5000.000 5000.000 .

---- EQU DEMANCON DEMAND CONSTRAINT

LOWER LEVEL UPPER MARGINAL

ATLANTA 3000.000 3000.000 +INF 400.000 CHICAGO 4000.000 4000.000 +INF 300.000 DENVER 1000.000 1000.000 +INF 200.000 Solution Report SOLVE MINCOST USING LP FROM LINE 59

---- VAR SHIPPED CARS SHIPPED FROM I TO J

LOWER LEVEL UPPER MARGINAL

DETROIT .ATLANTA . . +INF EPS DETROIT .CHICAGO . 3000.000 +INF . DETROIT .DENVER . . +INF 300.000 LOSANGEL.ATLANTA . 3000.000 +INF . LOSANGEL.CHICAGO . 1000.000 +INF . LOSANGEL.DENVER . 1000.000 +INF .

LOWER LEVEL UPPER MARGINAL

---- VAR TOTCOST -INF 2.0000E+6 +INF .

TOTCOST TOTAL COST OF SHIPPING CARS INPUT C:\COURSE\E480\EXP3.GMS OUTPUT C:\COURSE\E480\EXP3.LST

EXAMPLE 4: Model in Mathematical Notation

3 Maximize R = ' ri*Xi i = 1

3 Subject to: ' aij*Xi # bj i= 1,2,3 and j = 1,2,3 i = 1

Where: ri is the additional revenue collected and Xi is the number of returns audited of return type i, respectively aij is the time takes to audit return type i by auditor type j bj is the total hours of auditor type j available.

EXAMPLE 4: Model in GAMS Notation

$TITLE PETERSEN & LEWIS CHAPTER 8, similar to PROBLEM 11

* Constrained Tax Return Maximization: The tax office of a state * government wishes to determine the number of audits it should conduct

* on Individuals, Small Businesses, and Corporations, given the returns * resulting from the audits and given the constraints imposed by the

* maximum hours available of its CPA's, Bookkeepers and Investigators.

$OFFSYMLIST OFFSYMXREF OPTION NLP = MINOS5 OPTION RESLIM = 5400

SETS I ENTITIES TO BE AUDITED / INDIV, SMALLBUS, CORP / J PERSONNEL / CPA, BOOKEEP, INV /

*Data PARAMETERS TAXREV(I) ADDITIONAL TAX REVENUE COLLECTED PER AUDIT / INDIV 275 SMALLBUS 950 CORP 2200 /

HOURS(J) TOTAL HOURS AVAILABLE FOR PERSONNEL / CPA 300000 BOOKEEP 500000 INV 80000 /

TABLE TIME(I,J) TIME REQUIRED FOR EACH AUDIT BY EACH PERSONNEL

CPA BOOKEEP INV INDIV 5 5 0 SMALLBUS 8 10 10 CORP 30 15 24 ;

VARIABLES AUDIT(I) NUMBER OF AUDITS TOTTAX TOTAL ADDITIONAL TAX REVENUE;

POSITIVE VARIABLE AUDIT;

EQUATIONS OBJFUNC OBJECTIVE FUNCTION MAXHOURS(J) MAXIMUM PERSONNEL HOURS;

OBJFUNC .. TOTTAX =E= SUM(I, TAXREV(I)*AUDIT(I)); MAXHOURS(J) .. SUM(I, TIME(I,J) * AUDIT(I)) =L= HOURS(J);

MODEL MAXTAX / OBJFUNC, MAXHOURS /; SOLVE MAXTAX USING LP MAXIMIZING TOTTAX;

EXAMPLE 4: GAMS LIST FILE

GAMS 2.25.087 386/486 DOS 02/25/98 15:44:47 PAGE 1 PETERSEN & LEWIS CHAPTER 8, similar to PROBLEM 11

COMPILATION TIME = 0.050 SECONDS VERID MW2-25-087

Equation Listing SOLVE MAXTAX USING LP FROM LINE 59

---- OBJFUNC =E= OBJECTIVE FUNCTION

OBJFUNC.. - 275*AUDIT(INDIV) - 950*AUDIT(SMALLBUS) - 2200*AUDIT(CORP) + TOTTAX =E= 0 ; (LHS = 0)

---- MAXHOURS =L= MAXIMUM PERSONNEL HOURS

MAXHOURS(CPA).. 5*AUDIT(INDIV) + 8*AUDIT(SMALLBUS) + 30*AUDIT(CORP)=L=300000; (LHS= 0)

MAXHOURS(BOOKEEP).. 5*AUDIT(INDIV) + 10*AUDIT(SMALLBUS) + 15*AUDIT(CORP)=L= 500000; (LHS = 0)

MAXHOURS(INV).. 10*AUDIT(SMALLBUS) + 24*AUDIT(CORP) =L= 80000; (LHS = 0)

MODEL STATISTICS

BLOCKS OF EQUATIONS 2 SINGLE EQUATIONS 4 BLOCKS OF VARIABLES 2 SINGLE VARIABLES 4 NON ZERO ELEMENTS 12

GENERATION TIME = 0.050 SECONDS

EXECUTION TIME = 0.110

Solution Report SOLVE MAXTAX USING LP FROM LINE 59

S O L V E S U M M A R Y

MODEL MAXTAX OBJECTIVE TOTTAX TYPE LP DIRECTION MAXIMIZE SOLVER MINOS5 FROM LINE 59

**** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 1 OPTIMAL **** OBJECTIVE VALUE 20580000.0000

RESOURCE USAGE, LIMIT 0.280 5400.000 ITERATION COUNT, LIMIT 3 10000 M I N O S 5.3 (Nov 1990) Ver: 225-386-02 = = = = = B. A. Murtagh, University of New South Wales and P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright

EXIT -- OPTIMAL SOLUTION FOUND

LOWER LEVEL UPPER MARGINAL

---- EQU OBJFUNC . . . 1.000 OBJFUNC OBJECTIVE FUNCTION

---- EQU MAXHOURS MAXIMUM PERSONNEL HOURS

LOWER LEVEL UPPER MARGINAL CPA -INF 3.0000E+5 3.0000E+5 55.000 BOOKEEP -INF 3.1600E+5 5.0000E+5 . INV -INF 80000.000 80000.000 51.000

---- VAR AUDIT NUMBER OF AUDITS

LOWER LEVEL UPPER MARGINAL INDIV . 47200.000 +INF . SMALLBUS . 8000.000 +INF . CORP . . +INF -674.

Solution Report SOLVE MAXTAX USING LP FROM LINE 59

LOWER LEVEL UPPER MARGINAL

---- VAR TOTTAX -INF 2.0580E+7 +INF .

TOTTAX TOTAL AADITIONAL TAX REVENUE

**** REPORT SUMMARY : 0 NONOPT 0 INFEASIBLE 0 UNBOUNDED

EXECUTION TIME = 0.440 SECONDS VERID MW2-25-087

USER: U.S. Department of Agriculture G950224:1314CR-MW2 Washington, DC

**** FILE SUMMARY

INPUT C:\COURSE\E480\EXP4.GMS OUTPUT C:\COURSE\E480\EXP4.LST